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G = D5×D4.S3order 480 = 25·3·5

Direct product of D5 and D4.S3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×D4.S3, D20.7D6, Dic69D10, C60.7C23, Dic302C22, C3⋊C815D10, C36(D5×SD16), D4.7(S3×D5), (D4×D5).1S3, (C5×D4).1D6, D4.D151C2, (D5×Dic6)⋊1C2, (C3×D5)⋊2SD16, (C6×D5).61D4, (C4×D5).44D6, C6.140(D4×D5), C1510(C2×SD16), C153C85C22, (C3×D4).18D10, C30.169(C2×D4), C30.D42C2, C6.D202C2, C20.7(C22×S3), C12.7(C22×D5), (C3×Dic5).12D4, (C5×Dic6)⋊2C22, (C3×D20).3C22, (D4×C15).1C22, (D5×C12).3C22, D10.39(C3⋊D4), Dic5.12(C3⋊D4), (D5×C3⋊C8)⋊2C2, C4.7(C2×S3×D5), C52(C2×D4.S3), (C3×D4×D5).1C2, (C5×C3⋊C8)⋊5C22, (C5×D4.S3)⋊1C2, C2.22(D5×C3⋊D4), C10.43(C2×C3⋊D4), SmallGroup(480,559)

Series: Derived Chief Lower central Upper central

C1C60 — D5×D4.S3
C1C5C15C30C60D5×C12D5×Dic6 — D5×D4.S3
C15C30C60 — D5×D4.S3
C1C2C4D4

Generators and relations for D5×D4.S3
 G = < a,b,c,d,e,f | a5=b2=c4=d2=e3=1, f2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf-1=c-1, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >

Subgroups: 716 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22 [×5], C5, C6, C6 [×4], C8 [×2], C2×C4 [×2], D4, D4 [×2], Q8 [×3], C23, D5 [×2], D5, C10, C10, Dic3 [×2], C12, C12, C2×C6 [×5], C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10 [×3], C2×C10, C3⋊C8, C3⋊C8, Dic6, Dic6 [×2], C2×Dic3, C2×C12, C3×D4, C3×D4 [×2], C22×C6, C3×D5 [×2], C3×D5, C30, C30, C2×SD16, C52C8, C40, Dic10 [×2], C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C2×C3⋊C8, D4.S3, D4.S3 [×3], C2×Dic6, C6×D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5 [×3], C2×C30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, C2×D4.S3, C5×C3⋊C8, C153C8, D5×Dic3, C15⋊Q8, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, Dic30, D4×C15, D5×C2×C6, D5×SD16, D5×C3⋊C8, C30.D4, C6.D20, C5×D4.S3, D4.D15, D5×Dic6, C3×D4×D5, D5×D4.S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C2×SD16, C22×D5, D4.S3 [×2], C2×C3⋊D4, S3×D5, D4×D5, C2×D4.S3, C2×S3×D5, D5×SD16, D5×C3⋊D4, D5×D4.S3

Smallest permutation representation of D5×D4.S3
On 120 points
Generators in S120
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)(82 85)(83 84)(87 90)(88 89)(92 95)(93 94)(97 100)(98 99)(102 105)(103 104)(107 110)(108 109)(112 115)(113 114)(117 120)(118 119)
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)(61 91 76 106)(62 92 77 107)(63 93 78 108)(64 94 79 109)(65 95 80 110)(66 96 81 111)(67 97 82 112)(68 98 83 113)(69 99 84 114)(70 100 85 115)(71 101 86 116)(72 102 87 117)(73 103 88 118)(74 104 89 119)(75 105 90 120)
(1 49)(2 50)(3 46)(4 47)(5 48)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(1 79 19 64)(2 80 20 65)(3 76 16 61)(4 77 17 62)(5 78 18 63)(6 81 21 66)(7 82 22 67)(8 83 23 68)(9 84 24 69)(10 85 25 70)(11 86 26 71)(12 87 27 72)(13 88 28 73)(14 89 29 74)(15 90 30 75)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 111 51 96)(37 112 52 97)(38 113 53 98)(39 114 54 99)(40 115 55 100)(41 116 56 101)(42 117 57 102)(43 118 58 103)(44 119 59 104)(45 120 60 105)

G:=sub<Sym(120)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119), (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45)(61,91,76,106)(62,92,77,107)(63,93,78,108)(64,94,79,109)(65,95,80,110)(66,96,81,111)(67,97,82,112)(68,98,83,113)(69,99,84,114)(70,100,85,115)(71,101,86,116)(72,102,87,117)(73,103,88,118)(74,104,89,119)(75,105,90,120), (1,49)(2,50)(3,46)(4,47)(5,48)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,66,71)(62,67,72)(63,68,73)(64,69,74)(65,70,75)(76,81,86)(77,82,87)(78,83,88)(79,84,89)(80,85,90)(91,96,101)(92,97,102)(93,98,103)(94,99,104)(95,100,105)(106,111,116)(107,112,117)(108,113,118)(109,114,119)(110,115,120), (1,79,19,64)(2,80,20,65)(3,76,16,61)(4,77,17,62)(5,78,18,63)(6,81,21,66)(7,82,22,67)(8,83,23,68)(9,84,24,69)(10,85,25,70)(11,86,26,71)(12,87,27,72)(13,88,28,73)(14,89,29,74)(15,90,30,75)(31,106,46,91)(32,107,47,92)(33,108,48,93)(34,109,49,94)(35,110,50,95)(36,111,51,96)(37,112,52,97)(38,113,53,98)(39,114,54,99)(40,115,55,100)(41,116,56,101)(42,117,57,102)(43,118,58,103)(44,119,59,104)(45,120,60,105)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119), (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45)(61,91,76,106)(62,92,77,107)(63,93,78,108)(64,94,79,109)(65,95,80,110)(66,96,81,111)(67,97,82,112)(68,98,83,113)(69,99,84,114)(70,100,85,115)(71,101,86,116)(72,102,87,117)(73,103,88,118)(74,104,89,119)(75,105,90,120), (1,49)(2,50)(3,46)(4,47)(5,48)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,66,71)(62,67,72)(63,68,73)(64,69,74)(65,70,75)(76,81,86)(77,82,87)(78,83,88)(79,84,89)(80,85,90)(91,96,101)(92,97,102)(93,98,103)(94,99,104)(95,100,105)(106,111,116)(107,112,117)(108,113,118)(109,114,119)(110,115,120), (1,79,19,64)(2,80,20,65)(3,76,16,61)(4,77,17,62)(5,78,18,63)(6,81,21,66)(7,82,22,67)(8,83,23,68)(9,84,24,69)(10,85,25,70)(11,86,26,71)(12,87,27,72)(13,88,28,73)(14,89,29,74)(15,90,30,75)(31,106,46,91)(32,107,47,92)(33,108,48,93)(34,109,49,94)(35,110,50,95)(36,111,51,96)(37,112,52,97)(38,113,53,98)(39,114,54,99)(40,115,55,100)(41,116,56,101)(42,117,57,102)(43,118,58,103)(44,119,59,104)(45,120,60,105) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(42,45),(43,44),(47,50),(48,49),(52,55),(53,54),(57,60),(58,59),(62,65),(63,64),(67,70),(68,69),(72,75),(73,74),(77,80),(78,79),(82,85),(83,84),(87,90),(88,89),(92,95),(93,94),(97,100),(98,99),(102,105),(103,104),(107,110),(108,109),(112,115),(113,114),(117,120),(118,119)], [(1,49,19,34),(2,50,20,35),(3,46,16,31),(4,47,17,32),(5,48,18,33),(6,51,21,36),(7,52,22,37),(8,53,23,38),(9,54,24,39),(10,55,25,40),(11,56,26,41),(12,57,27,42),(13,58,28,43),(14,59,29,44),(15,60,30,45),(61,91,76,106),(62,92,77,107),(63,93,78,108),(64,94,79,109),(65,95,80,110),(66,96,81,111),(67,97,82,112),(68,98,83,113),(69,99,84,114),(70,100,85,115),(71,101,86,116),(72,102,87,117),(73,103,88,118),(74,104,89,119),(75,105,90,120)], [(1,49),(2,50),(3,46),(4,47),(5,48),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55),(61,66,71),(62,67,72),(63,68,73),(64,69,74),(65,70,75),(76,81,86),(77,82,87),(78,83,88),(79,84,89),(80,85,90),(91,96,101),(92,97,102),(93,98,103),(94,99,104),(95,100,105),(106,111,116),(107,112,117),(108,113,118),(109,114,119),(110,115,120)], [(1,79,19,64),(2,80,20,65),(3,76,16,61),(4,77,17,62),(5,78,18,63),(6,81,21,66),(7,82,22,67),(8,83,23,68),(9,84,24,69),(10,85,25,70),(11,86,26,71),(12,87,27,72),(13,88,28,73),(14,89,29,74),(15,90,30,75),(31,106,46,91),(32,107,47,92),(33,108,48,93),(34,109,49,94),(35,110,50,95),(36,111,51,96),(37,112,52,97),(38,113,53,98),(39,114,54,99),(40,115,55,100),(41,116,56,101),(42,117,57,102),(43,118,58,103),(44,119,59,104),(45,120,60,105)])

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C6D6E6F6G8A8B8C8D10A10B10C10D12A12B15A15B20A20B20C20D30A30B30C30D30E30F40A40B40C40D60A60B
order122222344445566666668888101010101212151520202020303030303030404040406060
size11455202210126022244101020206630302288420444424244488881212121288

48 irreducible representations

dim1111111122222222222224444448
type++++++++++++++++++-+++-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6SD16D10D10D10C3⋊D4C3⋊D4D4.S3S3×D5D4×D5C2×S3×D5D5×SD16D5×C3⋊D4D5×D4.S3
kernelD5×D4.S3D5×C3⋊C8C30.D4C6.D20C5×D4.S3D4.D15D5×Dic6C3×D4×D5D4×D5C3×Dic5C6×D5D4.S3C4×D5D20C5×D4C3×D5C3⋊C8Dic6C3×D4Dic5D10D5D4C6C4C3C2C1
# reps1111111111121114222222222442

Matrix representation of D5×D4.S3 in GL6(𝔽241)

100000
010000
001000
000100
00002401
000018852
,
100000
010000
001000
000100
00002400
00001881
,
010000
24000000
00240000
00024000
000010
000001
,
02400000
24000000
00240000
00133100
000010
000001
,
100000
010000
0015000
0022822500
000010
000001
,
192220000
2222220000
0021623700
001562500
000010
000001

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,188,0,0,0,0,1,52],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,188,0,0,0,0,0,1],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,240,0,0,0,0,240,0,0,0,0,0,0,0,240,133,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,228,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[19,222,0,0,0,0,222,222,0,0,0,0,0,0,216,156,0,0,0,0,237,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D5×D4.S3 in GAP, Magma, Sage, TeX

D_5\times D_4.S_3
% in TeX

G:=Group("D5xD4.S3");
// GroupNames label

G:=SmallGroup(480,559);
// by ID

G=gap.SmallGroup(480,559);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,135,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^4=d^2=e^3=1,f^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=f*c*f^-1=c^-1,c*e=e*c,d*e=e*d,f*d*f^-1=c*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽